4. Graphing Trigonometric Functions

Determine an equation for each graph.

Graph each function over a one-period interval.

y = -2 tan (¼ x)

Below is a graph of the function $y=\(\tan\]\left\)(bx\(\right\))$. Determine the value of b.

Graph each function over a two-period interval.

y = 1 + tan x

Determine the simplest form of an equation for each graph. Choose b > 0, and include no phase shifts. (Midpoints and quarter points are identified by dots.)

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Consider the function g(x) = -2 csc (4x + π). What is the domain of g? What is its range?

Graph each function over a one-period interval.

y = 1 - (1/2) csc (x - 3π/4)

Consider the following function from Example 5. Work these exercises in order.

y = -2 - cot (x - π/4)

Based on the answer in Exercise 58 and the fact that the cotangent function has period π, give the general form of the equations of the asymptotes of the graph of y = -2 - cot (x - π/4).

Let n represent any integer.

Graph each function over a one-period interval.

y = 2 + 3 sec (2x - π)

Graph each function over a two-period interval.

y = -1 + 2 tan x

A rotating beacon is located at point A, 4 m from a wall. The distance a is given by

a = 4 |sec 2πt|,

where t is time in seconds since the beacon started rotating. Find the value of a for each time t. Round to the nearest tenth if applicable.

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t = 1.24

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.

y = 2 sec(πx - 2π)

In Exercises 18–24, graph two full periods of the given tangent or cotangent function. y = −tan(x − π/4)

In Exercises 17–24, graph two periods of the given cotangent function. y = 3 cot(x + π/2)

Below is a graph of the function $y=\(\cot\[\left\)(bx+\(\frac{\pi}{2}\]\right\))$. Determine the value of b.